Birman–Solomyak Weyl-law conjecture for pseudodifferential operators on noncommutative tori

Let $P=P^*\in \Psi^{-m}(\mathbb{T}^n_\theta)$ with $m>0$ have principal symbol $\rho_{-m}(\xi)$, and set $p=nm^{-1}$. For a selfadjoint compact operator, write $\pm\lambda_j^{\pm}(P)$ for its positive and negative eigenvalues, listed in nonincreasing order and repeated according to multiplicity. Birman–Solomyak Weyl-law conjecture. One should have

$$\lim_{j\rightarrow \infty} j^{\frac{1}{p}}\lambda_j^{\pm}(P)=\left(\frac{1}{n}\int_{\mathbb{S}^{n-1}}\tau\left[\big(\rho_{-m}(\xi)_{\pm}\big)^p\right]d\xi\right)^{\frac{1}{p}}.$$

This conjectures the noncommutative-torus analogue of Birman–Solomyak’s Weyl law for negative-order pseudodifferential operators. It would give precise positive and negative eigenvalue asymptotics for such operators and, via the Birman–Schwinger principle, corresponding semiclassical asymptotics for fractional Schrödinger operators. The source presents it as an analogue that supersedes a conjecture from a prequel paper; no resolution is supplied here.